Question: Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{2p^3 + 24p^2 + 70p}{-5p^2 - 45p - 70}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {2p(p^2 + 12p + 35)} {-5(p^2 + 9p + 14)} $ $ x = -\dfrac{2p}{5} \cdot \dfrac{p^2 + 12p + 35}{p^2 + 9p + 14} $ Next factor the numerator and denominator. $ x = - \dfrac{2p}{5} \cdot \dfrac{(p + 7)(p + 5)}{(p + 7)(p + 2)}$ Assuming $p \neq -7$ , we can cancel the $p + 7$ $ x = - \dfrac{2p}{5} \cdot \dfrac{p + 5}{p + 2}$ Therefore: $ x = \dfrac{ -2p(p + 5)}{ 5(p + 2)}$, $p \neq -7$